
For \(\mu _k<0\)  (\(k=1,2\) ), there is no question that \(\mathbf {x}=0\)  is a fixed point (we may call it the “off” node); Eq. (REF ) reduces to its first term and \(\Phi (0)=0\) .

By suitably choosing the half planes—taking advantage of the relative sign in the numerator of the second term in Eq. (REF )—another fixed point \(\mathrm {N}:=(\nu _1,\nu _2)^T\)  (the “on” node) can be induced (through an inverse saddle-node bifurcation at the “on” location: in one variable, \(\frac{x^2}{2}-a\,(x-a)\,\theta (x-a)\)  resets the slope to zero at \(x=a\) ) if \((J\mathrm {N})_k>-\mu _k\) , \(k=1,2\)  (namely \(j_{11}\nu _1-j_{12}\nu _2>-\mu _1\) , \(j_{21}\nu _1-j_{22}\nu _2>-\mu _2\) ) with
\(\Phi (\mathrm {N})=\frac{j_{11}j_{21}\nu _1^2-2j_{12}j_{21}\nu _1\nu _2+j_{12}j_{22}\nu _2^2}{2\rho \tau _1\tau _2\det J}+\frac{j_{21}\nu _1\mu _1-j_{12}\nu _2\mu _2}{\rho \tau _1\tau _2\det J}.\) 
If \(\mu _1\)  is varied (as in [1]}, [2]}), equistability is achieved for
\(\mu _1=\frac{1}{2}\left[\frac{j_{12}\nu _2}{j_{21}\nu _1}(j_{21}\nu _1-j_{22}\nu _2+2\,\mu _2)-(j_{11}\nu _1-j_{12}\nu _2)\right].\) 
The intersection of the cores of the \(s_k(i_k)\)  (located at the solution \(\frac{j_{22}\mu _1-j_{12}\mu _2}{\det J},\frac{j_{21}\mu _1-j_{11}\mu _2}{\det J}\)  of \(J\mathbf {x}+\mathrm {M}=0\) ) is a (singular in this limit) saddle point. Figure REF  (b) illustrates this situation for the parameters quoted in the caption (the choice obeys to the fact that global stability makes condition \(j_{21}\nu _1-j_{22}\nu _2>-\mu _2\)  rather stringent).

As \(\mu _k\rightarrow 0\) , \(k=1,2\) , this saddle point moves toward the “off” node. After a (direct) saddle-node bifurcation, only the “on” node at \(x_k=\nu _k\)  remains, since conditions \((J\mathrm {N})_k>-\mu _k\) , \(k=1,2\)  are better satisfied, see Fig. REF  (a).

